Economic growth Notebook

We will use a dataset that was used in the paper: I just ran two million regressions by Sala-I-Martin and Model uncertainty in cross country growth regression by Fernandez et. al. The dataset has 41 possible explanatory variables with 72 countries. The data consists of 43 columns with the Country, y (economic growth in per capita GDP) and 41 possible predictor variables.

We now conduct model selection by exhaustive search. Note that we have 2^41 which is approximately 2 trillion possible regressions to run. The leaps package has some smart ways to search over this space by avoiding visiting parts of the space where the optimum cannot exist. It employs a branch and bound algorithm to search more efficiently. This would take a few minutes to run on a laptop. The model shows the bias-variance trade-off. We can also plot the variables identified using the plot command.

library(leaps)
model1 <- regsubsets(y ~ .,data = eg1, nvmax = 41)

plot(summary(model1)$rsq)

plot(summary(model1)$adjr2)

which.max(summary(model1)$adjr2)

[1] 27

plot(model1, scale = c("r2"))

plot(model1, scale = c("adjr2"))

We next use the forward stepwise selection method which runs much faster as should be expected. Note that the results are not identical to what we obtained with the exhaustive selection approach.

model2 <- regsubsets(y ~ ., data = eg1, nvmax = 41, method = "forward")

plot(summary(model2)$rsq)

plot(summary(model2)$adjr2)

plot(model2, scale = c("r2"))

plot(model2, scale = c("adjr2"))

Which variables are selected:

summary(model1)$which[27,]
summary(model2)$which[27,]

The results indicate that with model 1, we have
1) EquipInv,
2) Confucian, EquipInv,
3) Buddha, Confucian, EquipInv,
4) YrsOpen, Confucian, Protestants, EquipInv

while for model 2, we have
1) EquipInv,
2) Confucian, EquipInv,
3) Buddha, Confucian, EquipInv,
4) Buddha, Protestants, EquipInv, Confucian
and so on. The results are different from the two models.

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LASSO model: The results indicate that for variables such as EquipInv, YrsOpen and Confucian for many values of lambda, these occur while some other variables such as Abslat do not show up as often. Such results help illustrate the reliability of possible predictors for economic growth and can also cast doubts on the robustness of the results for certain variables which might be proposed as being correlated with growth.

model4$beta["EquipInv",]

        s0         s1         s2         s3         s4         s5         s6         s7         s8         s9        s10        s11 
0.00000000 0.02989896 0.05202483 0.06760673 0.08042867 0.09212352 0.10277946 0.11258867 0.12154191 0.12971174 0.13715581 0.14406270 
       s12        s13        s14        s15        s16        s17        s18        s19        s20        s21        s22        s23 
0.15040049 0.15383321 0.15867593 0.16258346 0.16444915 0.16555791 0.16661977 0.16734255 0.16832148 0.16948025 0.17126770 0.17293008 
       s24        s25        s26        s27        s28        s29        s30        s31        s32        s33        s34        s35 
0.17351524 0.17220683 0.16702669 0.16130824 0.15651196 0.15187566 0.14705527 0.14234527 0.13738774 0.13193100 0.12706968 0.12274067 
       s36        s37        s38        s39        s40        s41        s42        s43        s44        s45        s46        s47 
0.11929709 0.11510558 0.11298039 0.11149575 0.11069796 0.10979416 0.10902316 0.10822285 0.10826583 0.10805193 0.10978025 0.11122019 
       s48        s49        s50        s51        s52        s53        s54        s55        s56        s57        s58        s59 
0.11265440 0.11364832 0.11454541 0.11517577 0.11571790 0.11637044 0.11769020 0.11837771 0.11911269 0.11981383 0.12046601 0.12107996 
       s60        s61        s62        s63        s64        s65        s66        s67        s68        s69        s70        s71 
0.12163600 0.12214121 0.12259898 0.12310262 0.12363149 0.12418720 0.12446285 0.12470910 0.12491215 0.12509626 0.12525557 0.12540204 
       s72        s73        s74        s75        s76        s77        s78        s79        s80        s81        s82        s83 
0.12553559 0.12566579 0.12577917 0.12588798 0.12597718 0.12607745 0.12615294 0.12619169 0.12622857 0.12626553 0.12629465 0.12632607 
       s84        s85 
0.12635177 0.12637656